Yoshimichi Ueda
A review of our recent works on free product actions, partly in collaborations with Dimitri Shlyakhtenko and with Fumio Hiai, is given. We also work out type III theoretic subfactor analysis on the subfactors constructed in our previous works. This part can be read as supplementary appendix to the works.
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Morita equivalence of twisted inverse semigroup actions and discrete twisted partial actions are introduced. Morita equivalent actions have Morita equivalent crossed products.
In this paper we investigate the interrelation between the topological freedom of partial actions of discrete groups and faithful representations of partial crossed products.
We introduce the concept of crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We illustrate with examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.
Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its $C^{*}$-algebra is isomorphic to a crossed product.
In this paper, we define the notions of full pro-$C^{*}$-crossed product, respectively reduced pro-$C^{*}$-crossed product, of a pro-$C^{*}$-algebra $A[τ_Γ] $ by a strong bounded action $α$ of a locally compact group $G$ and investigate some their properties.
We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces. For this, we extend the theory of partial crossed products from groups to crossed modules; extend Takesaki-Takai duality to Abelian crossed modules; show that equivalent crossed modules have equivalent categories of actions on C*-algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules.
Let $G$ be a group and $X$ be a product system over a semigroup $P$. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Szép product $P\bowtie G$. We define a Zappa-Szép action of $G$ on $X$ to be a collection of functions on $X$ that are compatible with both actions from $P\bowtie G$ in a certain sense. Given a Zappa-Szép action of $G$ on $X$, we construct a new product system $X\bowtie G$ over $P\bowtie G$, called the Zappa-Szép product of $X$ by $G$. We then associate to $X\bowtie G$ several universal C*-algebras and prove their respective H…
We investigate the structures of crossed products of the Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups. We completely determine their ideal structures and compute the strong Connes spectra and K-groups.
In this work, for a given inverse semigroup we will define the crossed product of an inverse semigroup by a partial action. Also, we will associate to an inverse semigroup $G$ an inverse semigroup $S_G$, and we will prove that there is a correspondence between the covariant representation of $G$ and covariant representation of $S_G$. Finally, we will explore a connection between crossed products of an inverse semigroup actions and crossed products by partial actions of inverse semigroups.
We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.
We will study a faithful product type action of G_q that is the q-deformation of a connected semisimple compact Lie group G, and prove that such an action is induced from a minimal action of the maximal torus T of G_q. This enables us to classify product type actions of SU_q(2) up to conjugacy. We also compute the intrinsic group of G_{q,Ω}, the 2-cocycle deformation of G_q that is naturally associated with the quantum flag manifold T\backslash G_q.
Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.
We study actions of countable discrete amenable groups on unital separable simple nuclear Z-absorbing C*-algebras. Under a certain assumption on tracial states, which is automatically satisfied in the case of a unique tracial state, the crossed product is shown to absorb the Jiang-Su algebra Z tensorially.
We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
We investigate approximation properties for $C^*$-algebras and their crossed products by actions and coactions by locally compact groups. We show that Haagerup's approximation constant is preserved for crossed products by arbitrary amenable groups, and we show why this is not always true in the non-amenable case. We also examine similar questions for other forms of the approximation property.
We show that for any prime p and for any II$_1$ factor N there exist two $mathbb{Z}_{p^2}$-- actions on the free product factor $*_{1} ^{p}N$ that have the same outer invariant but are not outer conjugate. Therefore, in the case of free product factors, the outer invariant is not a complete invariant for outer conjugacy.
We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γis elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.
We correct the K-theory computation for ring C*-algebras associated with rings of integers in number fields.
We examine the ideal structure of crossed products B\rtimes G where B is a continuous-trace C*-algebra and the induced action of G on the spectrum of B is proper. In particular, we are able to obtain a concrete description of the topology on the spectrum of the crossed product in the cases where either G is discrete or G is a Lie group acting smoothly on the spectrum of B.
Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras s…
We compute the K-theory for C*-algebras naturally associated with rings of integers in number fields. The main ingredient is a duality theorem for arbitrary global fields. It allows us to identify the crossed product arising from affine transformations on the finite adeles with the analogous crossed product algebra over the infinite adele space.
For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,δ)$ and $(B,ε)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further de…
Takai duality and crossed product hosts in C*-actions
本文研究奇异 C*-作用下的交叉积宿主结构，验证 Takai 型对偶的有效性，通过 Landstad 代数建立与常规交叉积的联系，并给出基态存在性的充要条件及顺从群情形下完全与约化交叉积宿主的一致性。
Actions of locally compact groups and quantum groups on W*-ternary rings of operators are discussed and related crossed products introduced. The results generalise those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed point spaces for convolution operators generated by contractive, non-necessarily positive measures, both in the classical and in the quantum context.
The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.
We show that for any countable discrete maximally almost periodic group $G$ and any UHF algebra $A$, there exists a strongly outer product type action $α$ of $G$ on $A$. We also show the existence of countable discrete almost abelian group actions with a certain Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear $C^*$-algebras with tracial rank zero and a unique tracial state appearing as crossed products.
We use deformation-rigidity theory in von Neumann algebra framework to study probability measure preserving actions by wreath product groups. In particular, we single out large families of wreath products groups satisfying various type of orbit equivalence (OE) rigidity. For instance, we show that whenever $H$, $K$, $\G$, $\La$ are icc, property (T) groups such that $H\wr \G$ is measure equivalent to $K\wr \La$ then automatically $\G$ is measure equivalent to $\La$ and $H^{\G}$ is measure equivalent to $K^{\La}$. Rigidity results for von Neumann algebras arising from certain actions of such gr…
We define spectral freeness for actions of discrete groups on C*-algebras. We relate spectral freeness to other freeness conditions; an example result is that for an action of a finite group, spectral freeness is equivalent to strong pointwise outerness, and also to the condition that the strong Connes spectrum of the action of the integers generated by a nontrivial group element is always nontrivial. We then prove permanence results for reduced crossed products by exact spectrally free actions, for crossed products by arbitrary actions of the two element group, and for extensions, direct limi…
Tracially amenable actions and purely infinite crossed products
本文引入迹顺从作用概念并给出等价刻画，证明含自由群F2的群在简单、单位、Z-稳定C*-代数上的外迹顺从作用具有纯无限交叉积，并提供自由群在AF-代数上的具体例子。
The recently developed theory of partial actions of discrete groups on $C^*$-algebras is extended. A related concept of actions of inverse semigroups on $C^*$-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.
Selfless Reduced Crossed Product $C^{*}$-Algebras Arising from Almost Periodic Actions
该研究证明了一类特定殆周期作用对应的约化交叉积 $C^*$-代数的性质，并结合两种方法完成证明。
We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph whose each weak connected component is a rooted (possibly infinite) directed tree, and the labelling map is almost proper, which is proved to be a reasonable generalization of the earlier case, then …
We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Iz…
We define and study fibrations of topological groupoids. We interpret a groupoid fibration L->H with fibre G as an action of H on G by groupoid equivalences. Our main result shows that a crossed product for an action of L is isomorphic to an iterated crossed product first by G and then by H. Here groupoid action means a Fell bundle over the groupoid, and crossed product means the section C*-algebra.
Let d be a positive integer, let X be the Cantor set, and let Z^d act freely and minimally on X. We prove that the crossed product C* (Z^d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K_0 (C* (Z^d, X)) is determined by traces. We obtain the same conclusion for the C*-algebras of various kinds of aperiodic tilings.
We show that Bogoljubov actions of $\mathbb{R}$ on the free group factor $L(\mathbb{F}_{\infty})$ associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate.
Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $α$ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, α)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, α)$ also has stable rank one.
Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $α$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $α$ fixes any tracial state of $A$. Then tsr$(A \rtimes_αG) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.
We characterize supramenable groups in terms of existence of invariant probability measures for partial actions on compact Hausdorff spaces and existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a C*-algebra by a semi-direct product of groups as two iterated partial crossed products. We give conditions which ensure that such decomposition is possible.
We will show a centrally free action of an amenable rigid C$^*$-tensor category on a properly infinite von Neumann algebra has the Rohlin property. This enables us to prove the fullness of the crossed product of a full factor by minimal action of a compact group. As another application of the Rohlin property, we will show the uniqueness of centrally free cocycle actions of an amenable rigid C$^*$-tensor category on properly infinite von Neumann algebras.
This paper is motivated primarily by the question of when the maximal and reduced crossed products of a $G$-$C^*$-algebra agree (particularly inspired by results of Matsumura and Suzuki), and the relationships with various notions of amenability and injectivity. We give new connections between these notions. Key tools in this include the natural equivariant analogues of injectivity, and of Lance's weak expectation property: we also give complete characterizations of these equivariant properties, and some connections with injective envelopes in the sense of Hamana.
We consider a fixed free and proper action of a locally compact group $G$ on a space $T$, and actions $α:G\to \Aut A$ on $C^*$-algebras for which there is an equivariant embedding of $(C_0(T),\rt)$ in $(M(A),α)$. A recent theorem of Rieffel implies that $α$ is proper and saturated with respect to the subalgebra $C_0(T)AC_0(T)$ of $A$, so that his general theory of proper actions gives a Morita equivalence between $A\rtimes_{α,r}G$ and a generalised fixed-point algebra $A^α$. Here we investigate the functor $(A,α)\mapsto A^α$ and the naturality of Rieffel's Morita equivalence, focusing in parti…
Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
本文通过逆半群作用构造胚芽群胚及具体协变表示族，提出约化与本质交叉积C*-代数的新定义，避免双重换位子与局部乘子代数，并证明与已有定义同构。
The notions of Busby-Smith and Green type twisted actions are extended to discrete unital inverse semigroups. The connection between the two types, and the connection with twisted partial actions, are investigated. Decomposition theorems for the twisted crossed products are given.
Partial actions of discrete groups on $C^*$-algebras and the associated crossed products have been studied by Exel and McClanahan. We characterise these crossed products in terms of the spectral subspaces of the dual coaction, generalising and simplifying a theorem of Exel for single partial automorphisms. We then use this characterisation to identify the Cuntz algebras and the Toeplitz algebras of Nica as crossed products by partial actions.
We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the $C^*$-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the $C^*$-algebra of a quotient graph. Our main tool is Laca's dilation theory for endomorphic actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger $C^*$-algebra.
Given a partial action αof a group G on an associative algebra A we consider the crossed product A x_αG. Using the algebras of multipliers of ideals of A we prove that A x_αG is associative, provided that all ideals of A are idempotent. This generalizes a previous result on the associativity of A x_αG in the context of C*-algebras. We also give a criteria for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group al…
We study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.
We study groupoid actions on left cancellative small categories and their associated Zappa-Szép products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Szép products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation property are preserved by taking crossed products by actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a s…
Let G and H be two locally compact groups acting on a C*-algebra A by commuting actions. We construct an action on the crossed product AXG out of a unitary 2-cocycle u and the action of H on A. For A commutative, and free and proper actions of G and H, we show that if the roles of these two actions are reversed, and u is replaced by u*, then the corresponding generalized fixed-point algebras, in the sense of Rieffel, are strong-Morita equivalent. We apply this result to the computation of the K-theory of quantum Heisenberg manifolds.
Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.
We consider group actions of topological groups on C*-algebras of the types which occur in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a "crossed product host" in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action. We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also pre…
Banach algebra crossed products by inverse semigroup actions
本文提出了一种对逆半群作用下的巴拿赫代数交叉积进行普遍描述的方法，允许对交叉积的所有表示进行分解。
We show that indecomposable weak Kac algebras are free over their Cartan subalgebras and prove a duality theorem for their actions. Using this result, for any biconnected weak Kac algebra we construct a minimal action on the hyperfinite II_1 factor. The corresponding crossed product inclusion of II_1 factors has depth 2 and an integer index. Its first relative commutant is, in general, non-trivial, so we derive some arithmetic properties of weak Kac algebras from considering reduced subfactors.
For a twisted partial action Θof a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_ΘG is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_ΘG for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is …
In this work we present a new definition to the Partial Crossed Product by actions of inverse semigroups in a C^*-algebra, without using the covariant representations as Sieben did in [5]. Also we present an isomorphism between the partial crossed products by partial actions of a group G and the partial crossed product by actions of S(G), an inverse semigroup associated to G introduced by Exel in [2].
Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras
本文研究离散群上Fell丛在Hilbert丛上作用的张量积概念，探讨其与C*-对应、类Rieffel诱导过程的联系，并给出Fell丛的Fourier-Stieltjes代数相关结果。
On Fock covariance for product systems and the reduced Hao-Ng isomorphism problem by discrete actions
本文刻画了乘积系统的等变Fock协变单射表示，证明其与Nica协变性和Toeplitz表示一致，并据此解决了离散群广义规范作用的约化Hao-Ng同构问题。
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